(untitled) (European Journal of Pure and Applied Mathematics, 2021)
Given a connected graph G, we say that SV(G) is a cost effective dominating set in G if, each vertex in S is adjacent to at least as many vertices outside S as inside S and that every vertex outside S is adjacent to at least one vertex in S. The minimum cardinality of a cost effective dominating set is the cost effective domination number of G. The maximum cardinality of a cost effective dominating set is the upper cost effective domination number of G, and is denoted by γ_ce^+(G). A cost effective dominating set is said to be minimal if it does not contain a proper subset which is itself a cost effective dominating in G. The maximum cardinality of a minimal cost effective dominating set in a graph G is the minimal cost effective domination number of G, and is denoted by γ_mce(G). In this paper we provide bounds on upper cost effective domination number and minimal cost effective domination number of a connected graph G and characterized those graphs whose upper and minimal cost effective domination numbers are either 1, 2 or n-1. We also establish a Nordhaus-Gaddum type result for the introduced parameters and solve some realization problems.